Is it true that someone saying "forgive me" as he turns his back on me means he does not want to bear my sin on the Day of Judgment, and is there another ruling for that?

This statement is incorrect. There is no evidence that one person turning their back on another is a religiously prohibited act punishable on the Day of Judgment. Such actions should be considered in light of people's customs, which vary from country to country. What one group considers a fault, others do not see as such. Imam Al-Nawawi said: "And 'muru'ah' (chivalry/manliness) is to embody the character of one's peers in one's time and place." The matter varies with individuals, circumstances, and places.